The Find the difference quotient of $$ f(x)=x^{2}+1 $$; that is find $$ \frac{f(x+h)-f(x)}{h}, h
eq 0 $$. Be sure to simplify.

Question

The Find the difference quotient of $$ f(x)=x^{2}+1 $$; that is find $$ \frac{f(x+h)-f(x)}{h}, h 
eq 0 $$. Be sure to simplify.

UpStudy AI · Accepted Answer

1. Start with the function:
   $$
   f(x)=x^2+1.
   $$

2. Compute $$ f(x+h) $$ by replacing $$ x $$ with $$ x+h $$:
   $$
   f(x+h) = (x+h)^2 + 1 = x^2 + 2xh + h^2 + 1.
   $$

3. Form the difference $$ f(x+h) - f(x) $$:
   $$
   f(x+h) - f(x) = \left(x^2 + 2xh + h^2 + 1\right) - \left(x^2 + 1\right).
   $$
   Notice that the $$ x^2 $$ and $$ 1 $$ terms cancel:
   $$
   f(x+h) - f(x) = 2xh + h^2.
   $$

4. Divide the difference by $$ h $$ (with $$ h \neq 0 $$):
   $$
   \frac{f(x+h)-f(x)}{h} = \frac{2xh + h^2}{h}.
   $$

5. Factor $$ h $$ out of the numerator:
   $$
   \frac{2xh + h^2}{h} = \frac{h(2x + h)}{h}.
   $$
   Cancel the $$ h $$ (since $$ h \neq 0 $$):
   $$
   \frac{f(x+h)-f(x)}{h} = 2x + h.
   $$

The simplified difference quotient is:
$$
2x+h.
$$
The difference quotient simplifies to $$ 2x + h $$.

The Find the difference quotient of \( f(x)=x^{2}+1 \); that is find \( \frac{f(x+h)-f(x)}{h}, h \neq 0 \). Be sure to simplify.

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